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4.13.6 Example: Interactive Resolution Computation
In this example we compute the minimal free resolution of the ideal I
generated by the 2 by 2 minors of a catalecticant matrix, A, using the
interactive environment of the system. We define the ideal I, and
start the computation of its minimal free resolution using the
Hilbert-driven algorithm described in
A. Capani, G. De Dominicis, G. Niesi, L. Robbiano,
Computing Minimal Finite Free Resolutions,
J. Pure Appl. Algebra, Vol. 117--118, Pages 105--117, 1997.
example
Use R ::= Z/(32003)[z[0..3,0..3,0..3]]; -- set up the ring
A := Mat([ -- define the ideal
[z[3,0,0], z[2,1,0], z[2,0,1]],
[z[2,1,0], z[1,2,0], z[1,1,1]],
[z[2,0,1], z[1,1,1], z[1,0,2]],
[z[1,2,0], z[0,3,0], z[0,2,1]],
[z[1,1,1], z[0,2,1], z[0,1,2]],
[z[1,0,2], z[0,1,2], z[0,0,3]]
]);
I := Ideal(Minors(2,A));
GB.Start_Res(I); -- start interactive framework
GB.Steps(I,1000); -- first 1000 steps
GB.GetRes(I);
0 --> R^176(-5) --> R^189(-4) --> R^105(-3) --> R^27(-2)
-------------------------------
GB.ResReport(I);
--------------------------------------------------------------
Minimal Pairs, : 650
Groebner Pairs : 14
Minimal (Type S) : 636
H-Killed (Type S0) : 9
--------------------------------------------------------------
-------------------------------
GB.Complete(I); -- complete the calculation
GB.GetRes(I);
0 --> R(-9) --> R^27(-7) --> R^105(-6) --> R^189(-5) -->
R^189(-4) --> R^105(-3) --> R^27(-2)
-------------------------------
GB.ResReport(I);
--------------------------------------------------------------
Minimal Pairs, : 730
Groebner Pairs : 25
Minimal (Type S) : 705
Minimal (Type Smin) : 616
Minimal (Type S0) : 89
H-Killed (Type S0) : 78
Hard (Type S0) : 11
--------------------------------------------------------------
-------------------------------